Season the chicken: Set a large Dutch oven over medium heat, add 1 tablespoon of olive oil. Season the chicken all over with a big pinch of kosher salt and freshly ground black pepper. Add it to the Dutch oven.
Cook the chicken: Cook the chicken on one side for 12 minutes, until it develops a nice golden brown color. Then flip it over and cook it for another 8 to 12 minutes, letting color develop on both sides, and the bottom of the pan has a nice fond (brown bits.) The chicken should be cooked through when the thickest part reads 165 on an instant-read thermometer. (If you're concerned the chicken bits will burn, you can add a splash of stock and scrape the bits off the bottom of the pan then continue cooking the chicken.)
Let the chicken rest: Remove the chicken to a plate and set aside.
Cook the vegetables: To the Dutch oven, add the carrots, celery, onion, and potatoes. Stir, scraping any brown bits off the bottom of the pot. (You can add an additional splash of stock here to help scrape the pot, if needed.) Add the bay leaf, dried thyme, oregano, and red pepper flakes. Season with a big pinch of salt and freshly cracked pepper. Stir and let it sit undisturbed for about 15 minutes. Add the garlic. Stir and cook for another 5 minutes.
Finish the soup: Add the canned tomatoes, zucchini, and chicken stock and bring to a boil for about 15 to 20 minutes, until the potatoes have softened. While the potatoes soften, shred the chicken and add it back to the soup just until warmed through.
Garnish and serve: Finish with a 1/2 cup chopped fresh parsley and serve.